Unraveling Functions: Decomposing Composite Functions

Algebra 2 Grades High School 2:39 Video
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Lesson Description

Learn how to decompose composite functions, h(x) = f(g(x)), by identifying inner and outer functions. This lesson will help you reverse the composition process and find the original functions.

Video Resource

Decomposing Functions h(x) = f(g(x))

Mario's Math Tutoring

Duration: 2:39
Watch on YouTube

Key Concepts

  • Composition of Functions
  • Decomposition of Functions
  • Inner and Outer Functions

Learning Objectives

  • Students will be able to identify the inner and outer functions of a composite function.
  • Students will be able to decompose a given composite function into its original functions, f(x) and g(x).
  • Students will be able to verify the decomposition by composing the identified functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by briefly reviewing the concept of composite functions: f(g(x)). Explain that this lesson focuses on reversing the process – taking a composite function and breaking it down into its original components.
  • Video Viewing (10 mins)
    Play the 'Decomposing Functions h(x) = f(g(x))' video by Mario's Math Tutoring. Encourage students to take notes on the key concepts: inner function, outer function, and verification process.
  • Guided Practice (15 mins)
    Work through the examples from the video on the board, emphasizing the thought process behind identifying the inner and outer functions. Encourage students to participate and ask questions.
  • Independent Practice (15 mins)
    Provide students with practice problems where they decompose functions on their own. Circulate to provide assistance and answer questions.
  • Wrap-up (5 mins)
    Review the key concepts and address any remaining questions. Emphasize the importance of verifying the decomposition by composing the identified functions. Preview how this skill connects to more complex function analysis.

Interactive Exercises

  • Function Decomposition Challenge
    Present students with a series of increasingly complex composite functions. Challenge them to decompose the functions within a set time limit and verify their solutions. Offer bonus points for finding multiple valid decompositions.

Discussion Questions

  • Why is it important to understand how to decompose functions?
  • Is there always only one correct way to decompose a function? Explain.
  • How does the domain and range of the inner and outer functions affect the composite function?

Skills Developed

  • Analytical Thinking
  • Problem-Solving
  • Function Manipulation

Multiple Choice Questions

Question 1:

What is the first step in decomposing a composite function h(x) = f(g(x))?

Correct Answer: Identify the inner function g(x)

Question 2:

If h(x) = (x + 2)^2, which of the following could be f(x)?

Correct Answer: x^2

Question 3:

Which of the following is a valid way to check if you correctly decomposed a function?

Correct Answer: Compose f(x) and g(x) to see if you get h(x)

Question 4:

If g(x) = 3x - 1 and h(x) = √(3x - 1), what is f(x)?

Correct Answer: √x

Question 5:

In a composite function h(x) = f(g(x)), what is the 'outer' function?

Correct Answer: f(x)

Question 6:

If h(x) = 1/(x^2 + 1), which of the following is a possible decomposition?

Correct Answer: f(x) = 1/x, g(x) = x^2 + 1

Question 7:

Decomposing functions is the reverse process of:

Correct Answer: Composition

Question 8:

Which of the following functions CANNOT be easily decomposed?

Correct Answer: h(x) = x

Question 9:

If h(x) = |2x+3|, what is the outer function f(x)?

Correct Answer: f(x) = |x|

Question 10:

What does g(x) represent in the decomposition of h(x) = f(g(x))?

Correct Answer: The inner function

Fill in the Blank Questions

Question 1:

Decomposing functions involves identifying the _____ and outer functions.

Correct Answer: inner

Question 2:

The function h(x) = f(g(x)) is a _____ function.

Correct Answer: composite

Question 3:

To verify your decomposition, you should _____ f(x) and g(x).

Correct Answer: compose

Question 4:

If h(x) = √(x + 5), then a possible inner function g(x) is _____.

Correct Answer: x+5

Question 5:

The _____ function is 'plugged into' the outer function during composition.

Correct Answer: inner

Question 6:

If f(x) = x^3 and g(x) = 2x, then f(g(x)) = _____.

Correct Answer: 8x^3

Question 7:

If h(x)= sin(5x), the outer function f(x) is ______.

Correct Answer: sin(x)

Question 8:

The process of decomposition helps to _____ the structure of a complex function.

Correct Answer: reveal

Question 9:

Given h(x) = (x-1)^2, and f(x) = x^2, then g(x) is ______.

Correct Answer: x-1

Question 10:

If h(x) = 1/(x+7), the inner function g(x) could be ______.

Correct Answer: x+7

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together). A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain.

Teaching Materials

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